Question: Three points are chosen uniformly at random on a circle. What is the probability that no two of these points form an obtuse triangle with the circle's center?
Explanation: Let us call the circle's center $O$.  We first note that if $A$ and $B$ are points on the circle, then triangle $AOB$ is isosceles with $AO= BO$.  Therefore, if $AOB$ is an obtuse triangle, then the obtuse angle must be at $O$.  So $AOB$ is an obtuse triangle if and only if minor arc $AB$ has measure of more than $\pi/2$ ($90^\circ$).

Now, let the three randomly chosen points be $A_0$, $A_1$, and $A_2$.  Let $\theta$ be the measure of minor arc $A_0A_1$. Since $\theta$ is equally likely to be any value from 0 to $\pi$, the probability that it is less than $\pi/2$ is 1/2.

Now suppose that $\theta < \pi/2$.  For the problem's condition to hold, it is necessary and sufficient for point $A_2$ to lie within $\pi/2$ of both $A_0$ and $A_1$ along the circumference. As the diagram below shows, this is the same as saying that $A_2$ must lie along a particular arc of measure $\pi - \theta$.

[asy]
size(200);
defaultpen(.7);

pair O=(0,0), A=expi(4*pi/7), B=expi(3*pi/7);

draw(circle(O,1));

pair BB=rotate(90)*B;
pair AA=rotate(-90)*A;

pair LC= expi(5*pi/7), RC= expi(2*pi/7);

draw(O--BB..A..B..AA--O);

fill(O--BB..LC..A--cycle,gray(.8));
fill(O--A..(0,1)..B--cycle,gray(.6));
fill(O--B..RC..AA--cycle,gray(.8));

pair SA=1.15*A,SB=1.15*B,SBB=1.15*BB;
pair SAA=1.15*AA,SLC=1.15*LC,SRC=1.15*RC;

label("\(A_0\)",SA,N);
label("\(A_1\)",SB,N);

draw(SBB..SLC..SA,Arrows,Bars);
draw(SA..(0,1.15)..SB,Arrows);
draw(SB..SRC..SAA,Arrows,Bars);

label("\(\frac{\pi}{2}-\theta\)",SLC,NW);
label("\(\frac{\pi}{2}-\theta\)",SRC,NE);
label("\(\theta\)",(0,1.15),(0,1));
[/asy]

The probability of this occurrence is $\frac{\pi-\theta}{2\pi} = \frac{1}{2} - \frac{\theta}{2\pi}$, since $A_2$ is equally likely to go anywhere on the circle. Since the average value of $\theta$ between 0 and $\pi/2$ is $\pi/4$, it follows that the overall probability for $\theta < \pi/2$ is $\frac{1}{2} - \frac{\pi/4}{2\pi} = \frac{3}{8}$.

Since the probability that $\theta < \pi/2$ is 1/2, our final probability is $\frac{1}{2} \cdot \frac{3}{8} = \boxed{\frac{3}{16}}$.